The possibilities to run backups on a NDS are multiple, Supercard, M3, G6, DS-Xtreme, DSLink, EZ Flash, Magic Key, NinjaDS, R4 DS... and more I'm missing.
As Banned said, you'll find various models for each product, some will require to flash the BIOS, or will require to insert a Passkey in the NDS games slot, and some will only use that slot leaving the GBA slot empty (these won't run GBA games).

I bought myself a M3 Micro SD recently, it has the highest compatibiity with NDS and GBA backups, a high versatibility cos you can use up to 2gbs micro sd cards (not sure if a 4 gbs card will work too, mine is 2gbs), it's compatible with both NDS models, the original and the lite fitting in both perfectly but it's not the cheaper solution, the Supercard would be better if you want to save some money.